Question

A monatomic ideal gas is following the cyclic process $ABCA$. Then choose the incorrect option.

• A. Molar heat capacity for the process $AB$ is $\frac{R}{2}$
• B. Heat is rejected by the system in path $BC$
• C. Molar heat capacity for the process $BC$ is $\frac{2}{3}R$
• D. Work done by the system in the process $CA$ is $\frac{2U_0}{3}ln4$

Answer 1

Answer: (C)

For path $AB$
$TV=$ constant $\Rightarrow P{V}^2=$ constant $...(i)$
For polytropic process $P{V}^m=$ constant $...(ii)$
On comparing eqn. $(i)$ and $(ii)$ $m=2$ The molar heat capacity is given as
$C=\frac{R}{\gamma -1}-\frac{R}{m-1},\gamma =\frac{5}{3}$
$C=\frac{3R}{2}-R=\frac{R}{2}$
$C=\frac{R}{\frac{5}{3}-1}-\frac{R}{2-1}$
$=\frac{3}{2}R-R=\frac{R}{2}$
So option $(a)$ is correct.
For $BC$ it is an isochoric process and temperature is
decreasing so option $(b)$ is correct.
$C_V=\frac{3}{2}R;$ so option $(c)$ is wrong.
' For process $CA$ (isothermal)
$W=nRTln\left(\frac{V_A}{V_C}\right)$
$W=\frac{2}{3}\left(\frac{3}{2}nRT\right)ln\left(4\right)$ (AS density at $C$ is $4{\rho }_0)$
$=\frac{2}{3}{U}_{0}ln\left(4\right)$
So, option $(d)$ is correct.